Imaginary number

ImprimirCitar

In mathematics, particularly in algebra, a imaginary number is a complex number whose real part is equal to zero. For example, ${displaystyle 3i }$ is an imaginary number, as well as $i$ or $-i$ They are also imaginary numbers. In general an imaginary number is shape ${displaystyle z=y,i}$ Where $y$ It's a real number.

Definition

Imaginary numbers can be expressed as the product of a real number times the imaginary unit i, where the letter i denotes the root square of -1, that is:

{displaystyle z=y,i}

In square root imaginary numbers are the remainder of a negative root; that is, i: the square root of -1, -2, -3, -4, etc.

Appearance and uses

History and origin

The genre of complex/imaginary numbers was invented by Raffaelle Bombelli, a 16th-century Italian mathematician and engineer. The term imaginary numbers was created by René Descartes, in his treatise Geometry, in opposition to Bombelli's theories.

It was in 1777 when Leonhard Euler gave him $sqrt{-1}$ the name of iby imaginary, in a derogatory way, implying that he had no real existence. Gottfried Leibniz, in the 17th century, said $sqrt{-1}$ "a kind of amphibian between being and nothing".

In electrical engineering and related fields, the imaginary unit is often denoted by j to avoid confusion with the strength of an electric current, traditionally denoted by i.

Timeline

Year	Development
1572	Rafael Bombelli performs calculations using imaginary numbers.
1777	Leonhard Euler uses the symbol “i” to represent the square root of -1.
1811	Jean-Robert Argand creates the graphical representation of the complex plan also known as plane of Argand

Other representations

As an orderly pair of real numbers, it denotes z = (0, and)
Trigonometrically, z = r•cos(π/2) + r•sen(π/2)•i, where r is any real number.

Geometric interpretation

The product per

i

performs rotations of 90 degrees

Geometrically, the imaginary numbers are represented on the vertical axis of the complex plane and therefore perpendicular to the actual axis that is horizontal, the only element they share is the zero, since ${displaystyle 0=0i}$ . This vertical axis is called the " Imaginary Axis" and is denoted as $imathbb{R}$ , $mathbb{I}$ or simply $Im$ . In this representation you have to:

a multiplication by –1 corresponds to a rotation of 180 degrees on the origin.

A multiplication by $i$ corresponds to a rotation of 90 degrees in the "positive" sense (in the anti-horn sense), and the square of the equation $i^2 = -1$ can be interpreted as performing two rotations of 90 degrees on the origin, equivalent to a rotation of 180 degrees, $-1$ .

A 90-degree rotation in the "negative" direction (temporary sense) also satisfies this interpretation, as $-i$ is also a solution to the equation $x^2 = -1$ .

In general, multiplying by a complex number is the same as being rotated around the origin by the complex number argument, followed by scaling by its magnitude.

Properties

$vdots$
${displaystyle i^{-3}=;;;i}$
${displaystyle i^{-2}=-1}$
${displaystyle i^{-1}=-i}$
${displaystyle i^{0};;=;;;1}$
${displaystyle i^{1};;=;;;i}$
${displaystyle i^{2};;=-1}$
${displaystyle i^{3};;=-i}$
${displaystyle i^{4};;=;;;1}$
${displaystyle i^{5};;=;;;i}$
${displaystyle i^{6};;=-1}$
$vdots$
${displaystyle i^{n};;=;;;i^{noperatorname {mod} 4}}$ (mod represents the residue)
$vdots$

Every imaginary number can be written as $ib$ where $b$ It's a real number. $i$ It's imaginary unity.

Demonstration
Like $i^2 = -1$ you have to: ${displaystyle (bi)^{2}=b^{2}i^{2}=b^{2}(-1)=-b^{2};}$ which is a real number. Sea ${displaystyle -b}$ a negative real number has to: ${displaystyle {sqrt {-b}}={sqrt {(-1)b}}}$ ${displaystyle ={sqrt {-1}}{sqrt {b}}}$ ${displaystyle =i;{sqrt {b}}}$ ${displaystyle ={sqrt {b}};i.}$

Each complex number can be uniquely written as a sum of a real number and an imaginary number, like this:

$a + bi ,!$

The imaginary number i is also called the imaginary constant.

These numbers extend the set of real numbers $R$ to the complex numbers set $mathbb{C}$ .

On the other hand, we cannot assume that the imaginary numbers have the property, like the actual numbers, of being able to be sorted according to its value. I mean, it's right to say that $0}" xmlns="http://www.w3.org/1998/Math/MathML">1▪0{displaystyle 1 PHP0}0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efdd11636ca6f235c5057ead13e53ac89a9ba25c" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;"/>$ and what $<math alttext="{displaystyle -1- - 1.0{displaystyle -1 tax0} <img alt="-1$ ; this is because $0}" xmlns="http://www.w3.org/1998/Math/MathML">1− − 0▪0{displaystyle 1-0/20050}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c6b23e37e4f85d2baf3216d56e39ad887275c" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;"/>$ and $<math alttext="{displaystyle -1-0- - 1- - 0.0{displaystyle -1-0 tax0} <img alt="{displaystyle -1-0$ . This rule does not apply to imaginary numbers, due to a simple demonstration:

Let us remember that in the real numbers, the product of two real numbers, a and b, where both are greater than zero, is equal to a greater number than zero. For example, it is fair to say $0}" xmlns="http://www.w3.org/1998/Math/MathML">a=2▪0{displaystyle a=2/20050} 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efc4b42b33031b01f66c87124118c22d13750f88" style="vertical-align: -0.338ex; width:9.752ex; height:2.176ex;"/>$ , $0}" xmlns="http://www.w3.org/1998/Math/MathML">b=3▪0{displaystyle b=36.0000} 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3a1cd7a547496e59f3ba6c69088f8cac96c443" style="vertical-align: -0.338ex; width:9.519ex; height:2.176ex;"/>$ , therefore, $0}" xmlns="http://www.w3.org/1998/Math/MathML">(a)(b)=c▪0{displaystyle (a)(b)=c/2005} 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e02ed9f1c358452c705be9369c24cf1a9e54c8" style="vertical-align: -0.838ex; width:14.212ex; height:2.843ex;"/>$ So we have to $(2)(3) = 6$ and obviously $0}" xmlns="http://www.w3.org/1998/Math/MathML">6▪0{displaystyle 6 HCFC}0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d390c6e234108ec7fc61040f5dc5854f00602350" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;"/>$ .

On the other hand, suppose that $0}" xmlns="http://www.w3.org/1998/Math/MathML">i▪0{displaystyle i vocabulary0}0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f49f2878fd68a89c3da37eb537198e887cf0293" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;"/>$ So we have to $0}" xmlns="http://www.w3.org/1998/Math/MathML">− − 1=(i)(i)▪0{displaystyle -1=(i)(i) plan0} 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234e07aa3e5aaa84570c99c9487c11b5b31c428d" style="vertical-align: -0.838ex; width:15.553ex; height:2.843ex;"/>$ Which is obviously false.

And likewise, let's make the wrong assumption that $<math alttext="{displaystyle ii.0{displaystyle i vis0} <img alt="i$ But if we multiply by $-1$ We have to stay $0}" xmlns="http://www.w3.org/1998/Math/MathML">− − i▪0{displaystyle -i한0} 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a1e52954f1db635d971e7896a1ed17a170001a" style="vertical-align: -0.505ex; width:6.872ex; height:2.343ex;"/>$ . So we have to $0}" xmlns="http://www.w3.org/1998/Math/MathML">− − 1=(− − i)(− − i)▪0{displaystyle -1=(-i)(-i)/2005 0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a9fcc073e341d8d592062dc5562598eafb63b6" style="vertical-align: -0.838ex; width:19.17ex; height:2.843ex;"/>$ . What it is, equally that the previous assumption, totally false.

We will conclude that this assumption and any other of trying to give an ordinal value to imaginary numbers is completely wrong.

Applications

The imaginary unit can be used to formally obtain the square roots of negative numbers.
Likewise the square roots of an imaginary number are complex numbers, where one of them is of the form k (cos π/4 + i senπ/4) where k is a real number any.

In quantum physics the imaginary unit allows to simplify the mathematical description of the variable quantum states in time.
In circuit and current theory the imaginary unit is used to represent certain quantities as fastors, which allows a simpler algebraic treatment of such magnitudes.

Contenido relacionado

Más resultados...